Cream butter and sugar: Beat the softened butter and granulated sugar together until pale and fluffy, about 2 minutes. This airy texture helps the cookies stay tender after baking.
Add egg and vanilla: Beat in the egg and vanilla extract until fully combined and smooth. Scrape down the bowl so no streaks of butter remain.
Mix in dry ingredients: Whisk the flour, baking powder, and salt together, then add to the wet mixture in two additions, mixing just until a soft dough forms. Overmixing at this stage makes the cookies tough.
Chill the dough: Divide the dough into two flat disks, wrap in plastic, and refrigerate for at least 1 hour. Chilled dough rolls evenly and holds the daisy shape better.
Roll and cut daisies: On a floured surface, roll the dough to a 1/4-inch thickness and cut daisy shapes with a flower cutter. Gather scraps, re-roll once, and cut more shapes.
Chill the cutouts: Place the cut cookies on a parchment-lined baking sheet and chill for 10 minutes. This second chill keeps the petals from spreading in the oven.
Bake until edges are dry: Bake at 350°F for 9 to 11 minutes, until the edges look dry and matte and the centers are still pale. Avoid letting the edges turn golden brown.
Cool completely: Let the cookies rest on the baking sheet for 5 minutes, then transfer to a wire rack to cool fully, about 30 minutes. Icing warm cookies will cause the icing to melt and run.
Make the royal icing: Whisk the powdered sugar and meringue powder together, then add water one tablespoon at a time until the icing is thick but pourable. Divide into two bowls, tinting one white and one yellow.
Outline the petals: Using a piping bag fitted with a small round tip, outline each petal shape on the cooled cookies with white icing. Let the outline set for 5 minutes before flooding.
Flood the petals: Thin a portion of the white icing with a few drops of water until it flows like honey, then fill in each petal outline. Use a toothpick to pop any air bubbles and smooth the surface.
Pipe the yellow centers: Once the white icing has set for about 10 minutes, pipe a small circle of yellow icing in the center of each flower. Let the icing dry uncovered for at least 4 hours before stacking or storing.